Boolean prime ideal theorem

Let A be a Boolean algebraMathworldPlanetmath. Recall that an ideal I of A if it is closed under , and for any aI and bA, abI. I is proper if IA and non-trivial if I(0), and I is prime if it is proper, and, given abI, either aI or bI.

Theorem 1 (Boolean prime ideal theorem).

Every Boolean algebra contains a prime idealPlanetmathPlanetmathPlanetmath.


Let A be a Boolean algebra. If A is trivial (the two-element algebra), then (0) is the prime ideal we want. Otherwise, pick aA, where 0a1, and let be the trivial ideal. By Birkhoff’s prime ideal theorem for distributive lattices, A, considered as a distributive latticeMathworldPlanetmath, has a prime ideal P (containing (0) obviously) such that aP. Then P is also a prime ideal of A considered as a Boolean algebra. ∎

There are several equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath versions of the Boolean prime ideal theorem, some are listed below:

  1. 1.

    Every Boolean algebra has a prime ideal.

  2. 2.

    Every ideal in a Boolean algebra can be enlarged to a prime ideal.

  3. 3.

    Given a set S in a Boolean algebra A, and an ideal I disjoint from S, then there is a prime ideal P containing I and disjoint from S.

  4. 4.

    An ideal and a filter in a Boolean algebra, disjoint from one another, can be enlarged to an ideal and a filter that are complementPlanetmathPlanetmath (as sets) of one another.


  1. 1.

    Because the Boolean prime ideal theorem has been extensively studied, it is often abbreviated in the literature as BPI. Since the prime ideal theorem for distributive lattices uses the axiom of choiceMathworldPlanetmath, ZF+AC implies BPI. However, there are models of ZF+BPI where AC fails.

  2. 2.

    It can be shown (see John Bell’s online article that BPI is equivalent, under ZF, to some of the well known theorems in mathematics:


  • 1 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
  • 2 T. J. Jech, The Axiom of Choice, North-Holland Pub. Co., Amsterdam, (1973).
  • 3 R. Sikorski, Boolean Algebras, 2nd Edition, Springer-Verlag, New York (1964).
Title Boolean prime ideal theorem
Canonical name BooleanPrimeIdealTheorem
Date of creation 2013-03-22 18:45:52
Last modified on 2013-03-22 18:45:52
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Theorem
Classification msc 06E05
Classification msc 03G05
Classification msc 03E25
Synonym BPI